Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 13.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{2} + 29 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 19 a + 10 + \left(10 a + 22\right)\cdot 31 + \left(a + 11\right)\cdot 31^{2} + \left(30 a + 16\right)\cdot 31^{3} + \left(27 a + 15\right)\cdot 31^{4} + \left(10 a + 6\right)\cdot 31^{5} + \left(13 a + 5\right)\cdot 31^{6} + \left(29 a + 4\right)\cdot 31^{7} + \left(16 a + 24\right)\cdot 31^{8} + \left(5 a + 3\right)\cdot 31^{9} + \left(12 a + 12\right)\cdot 31^{10} + \left(14 a + 18\right)\cdot 31^{11} + \left(29 a + 18\right)\cdot 31^{12} +O\left(31^{ 13 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 26 + 3\cdot 31 + 12\cdot 31^{2} + 30\cdot 31^{3} + 14\cdot 31^{4} + 11\cdot 31^{5} + 13\cdot 31^{6} + 4\cdot 31^{7} + 10\cdot 31^{8} + 13\cdot 31^{9} + 15\cdot 31^{10} + 24\cdot 31^{11} + 15\cdot 31^{12} +O\left(31^{ 13 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 12 a + 17 + \left(20 a + 24\right)\cdot 31 + \left(29 a + 3\right)\cdot 31^{2} + 13\cdot 31^{3} + \left(3 a + 10\right)\cdot 31^{4} + 20 a\cdot 31^{5} + \left(17 a + 21\right)\cdot 31^{6} + \left(a + 18\right)\cdot 31^{7} + \left(14 a + 28\right)\cdot 31^{8} + \left(25 a + 28\right)\cdot 31^{9} + \left(18 a + 30\right)\cdot 31^{10} + \left(16 a + 3\right)\cdot 31^{11} + \left(a + 1\right)\cdot 31^{12} +O\left(31^{ 13 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 4 + 15\cdot 31 + 15\cdot 31^{2} + 31^{3} + 5\cdot 31^{4} + 24\cdot 31^{5} + 4\cdot 31^{6} + 8\cdot 31^{7} + 9\cdot 31^{8} + 29\cdot 31^{9} + 18\cdot 31^{10} + 8\cdot 31^{11} + 11\cdot 31^{12} +O\left(31^{ 13 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 2 a + 16 + \left(20 a + 25\right)\cdot 31 + \left(22 a + 27\right)\cdot 31^{2} + \left(a + 9\right)\cdot 31^{3} + \left(30 a + 25\right)\cdot 31^{4} + \left(5 a + 18\right)\cdot 31^{5} + \left(13 a + 29\right)\cdot 31^{6} + 3 a\cdot 31^{7} + \left(6 a + 6\right)\cdot 31^{8} + 27 a\cdot 31^{9} + \left(24 a + 12\right)\cdot 31^{10} + \left(14 a + 16\right)\cdot 31^{11} + \left(12 a + 2\right)\cdot 31^{12} +O\left(31^{ 13 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 29 a + 20 + \left(10 a + 1\right)\cdot 31 + \left(8 a + 22\right)\cdot 31^{2} + \left(29 a + 21\right)\cdot 31^{3} + 21\cdot 31^{4} + 25 a\cdot 31^{5} + \left(17 a + 19\right)\cdot 31^{6} + \left(27 a + 25\right)\cdot 31^{7} + \left(24 a + 14\right)\cdot 31^{8} + \left(3 a + 17\right)\cdot 31^{9} + \left(6 a + 3\right)\cdot 31^{10} + \left(16 a + 21\right)\cdot 31^{11} + \left(18 a + 12\right)\cdot 31^{12} +O\left(31^{ 13 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,4)$ |
| $(1,2,3,5)(4,6)$ |
| $(2,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $9$ |
$2$ |
$(1,3)(2,5)$ |
$0$ |
| $4$ |
$3$ |
$(1,3,4)$ |
$1$ |
| $4$ |
$3$ |
$(1,3,4)(2,5,6)$ |
$-2$ |
| $9$ |
$4$ |
$(1,2,3,5)(4,6)$ |
$0$ |
| $9$ |
$4$ |
$(1,5,3,2)(4,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
